NCERT Solutions

We need to calculate the magnetic torque on a circular coil and determine if shape affects the result.

Concept: Magnetic torque

The magnetic torque $\tau$ acting on a planar current-carrying coil placed in a uniform magnetic field is determined by the magnetic dipole moment and the field strength. It is given by the formula:

$$\tau = N I A B \sin \theta$$

where $N$ is the number of turns, $I$ is the current, $A$ is the area of the coil, $B$ is the magnetic field magnitude, and $\theta$ is the angle between the magnetic field and the normal vector to the coil's plane.

Given:

• Number of turns: $N = 30$
• Radius of the coil: $r = 8.0\,\text{cm} = 0.08\,\text{m}$
• Current: $I = 6.0\,\text{A}$
• Magnetic field magnitude: $B = 1.0\,\text{T}$
• Angle with the normal: $\theta = 60^{\circ}$

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Solving:

1. Calculate the area of the circular coil:

The area $A$ of a circle is given by:

$$A = \pi r^2$$

Substituting the given radius:

$$A = \pi \cdot (0.08\,\text{m})^2 = 0.0064\pi\,\text{m}^2$$

2. Calculate the magnitude of the magnetic torque:

Using the torque formula:

$$\begin{aligned} \tau &= N I A B \sin \theta \\ &= 30 \cdot 6.0\,\text{A} \cdot (0.0064\pi\,\text{m}^2) \cdot 1.0\,\text{T} \cdot \sin(60^{\circ}) \\ &= 180 \cdot 0.0064\pi \cdot \frac{\sqrt{3}}{2} \\ &= 90 \cdot 0.0064\pi \cdot \sqrt{3} \\ &= 0.576\pi\sqrt{3}\,\text{N}\cdotp\text{m} \end{aligned}$$

Evaluating the numerical value:

$$\tau \approx 3.1342\,\text{N}\cdotp\text{m}$$

3. Effect of coil shape (Part b):

The expression for torque $\tau = N I A B \sin \theta$ depends only on the area $A$ enclosed by the loop and not on its specific geometric shape. Therefore, if the circular coil is replaced by an irregular planar coil enclosing the same area, the magnitude of the torque remains unchanged.

Result:

(a) The magnitude of the counter torque required is approximately:

$$3.13\,\text{N}\cdotp\text{m}$$

(b) The answer would not change because the torque depends only on the area of the planar coil.

$$3.13\,\text{N}\cdotp\text{m}$$